You can be a cybered mage. Every full point of Essence lost (round up) lowers your current and maximum magic rating by an equal amount. You do lose your magical abilities if your current magic drops to 0.
Building an effective cybered mage is another story. It's quite possible and even arguable to be one. But it requires careful building and planning. Thankfully for you, cyberlimbs are totally part of the plan.
First off, you're going to stay a mage. You really paid way too much to be anything else. Therefore we can't really bomb your magic too much. Magic 3 is what I consider the cut off. Force 6 spells and spirits are quite effective whereas force 4 tends to be useful to meh. Since pointwise you'll "start" with a magic of 5, that gives you up to 2 essence worth of gear to lose. I suggest you follow a logic tradition for reasons that will become clear.
I suggest you be a formori. Metagentic Improvement (Body) combos well with cyberlimbs and arcane arrestor is one of the few effective anti-magic defenses around. However, not all groups use metavarients so I'll stay with a normal troll.
Now for qualities, you'll want Magician (duh), Mentor Spirit (effective for its price and neat), and Biocompatibility (Cyberware). This will use up 25-30 of your positive qualities depending on if you're SURGEd or not (I'll assume not here as well). The biocompatibility is only going to give you .2 essence, but making a cybermage is about using every part of your essence. I'm going to assume the cost of the positive qualities and negative qualities balance out so they won't factor into my calculations.
When designing the cyberlimbs, you'll run headfirst to one of the weirder aspects of cyberlimbs as pointed out before. Trolls and cyberlimbs have an odd relationship. While cyberlimbs on trolls have high customization maximums, the limbs still start at 3. Since each level of customization gives you +1 availability, you can rapidly hit the availability cap of 12 and you can't really afford to blow a restricted gear on them. But we can adjust.
Umaro is right that you'll likely want 2 legs or 1 arm and leg for the extra capacity. I assume you want the look of two limbs and extra 5 capacity is a small price to pay.
How you want to do the arms will depend on how your group averages cyberlimbs and meat limbs. RAW it's divided your over 2 arms, 2 legs, torso and head. However, most people, including the person who wrote the example in the book, do it differently. The most common way I've heard is to treat the head as a partial limb (though they don't call it that) so it's averaged over 2 arms, 2 legs, and the torso. So that's what I'm going with.
You want to get your body up. I'm assuming you're going with a Body of 5 (min Troll) to save on points. That means your meat limbs will add to 15. So let's get your arms customized to 9 and 10. This will the +3 body enhancement gives you 12 and 13 body arms. This gives an average body of 8 [(12+13+5+5+5)/5]. With the left over availability, we can get the following limbs.
Primary Limb || Secondary Limb
Body 9 (12) || 10(13)
Agility 4 (7) || 3 (5)
Strength 3 (5) || 3 (5)
Capacity Used: 12/15 || 11/15
Accessories: Body Enhancement 3, Agility Enhancement 3, Armor Enhancement 2, Strength Enhancement 2 || As other limb but Agility 2
Assuming you have the troll minimum attributes of Body 5, Agility 1, Strength 5, then gives you an average of Body 8, Agility 3, Strength 5. Pretty nice. And if you find yourself in a high background count zone, you can shoot dudes with your agility 7 cyberlimb. You'll want to increase your armor to 3 once play begins. You can play around with the limbs some and add stuff like gyromounts or whatever once play begins or use the last point of customization to up strength or whatever.
Now for the essence. You'll have to get your cyberlimbs alphaware. This means the limbs will cost 55900Y and 55400Y for 116300Y total. This combined with your force 3 sustaining focus (for increase reflexes) for 30,000Y and force 4 power focus (good buy for you right now since you have other uses for your karma and cash) for 100,000 is 241,300. That gives you 8700Y left over for lifestyle, armor, and misc stuff.
In exchange for this, you have .6 essence left to play with between alphaware and biocompatibility discounts. Once play begins, you want to buy two alphaware armor enhancement 3 for 1200Y each. With the remaining essence you want to fill it with bioware. Since you're a logic tradition, buy cerebral boosters 3 (30,000Y) right away for drain . After that, you can get other stuff with the remaining .6 bioware essence (due to the halving). I like a pain editor or trauma dampener for more toughness, but your tastes can vary. You'll want to spend your karma on normal mage stuff like buying up magic (costs you only 20 karma for that 4 point) or getting an ally spirit or whatever.
You're not the best mage ever. But you're nearly as good as other mages since you're only down 2 dice compared to them, however you're a hell of a lot tougher (6 armor, likely body score, and 2 physical condition boxes) and you have pretty decent physical statistics. You'll also likely have a slightly better drain after a bit of cash (logic 7 + willpower 5 versus soft caps mental attribute 5 + willpower 5).
Edit: I suggest billing yourself as a combat mage. You're tough and with a bit of skill decent with a gun in a pinch. Get a combat mentor spirit, specialize in combat spells and rock the world with force 6 stunbolts/stunballs.
